Ariah has a hot air balloon. When full, the balloon is a sphere with a radius of $9$ meters. She fills it with hot air whose density is $0.9$ kilograms per cubic meter. Since this is less dense than the air in the atmosphere, the balloon is carried upwards. What is the weight of the air that Ariah needs to fill the balloon? Round your answer, if necessary, to the nearest integer.
Explanation: This is a density word problem. To solve it, we can use the following equation, which is the volume definition of density: ${\text{Density}}=\dfrac{{\text{Total quantity}}}{{\text{Volume}}}$ What do we know? The ${\text{density}}$ of the hot air is ${0.9}$ kilograms per cubic meter. The balloon's radius is $9$ meters (we can use this to find the ${\text{volume}}$ ). What do we need to find? The weight of air in the balloon, which is the ${\text{total quantity}}$. The ${\text{volume}}$ of the spherical balloon is $\dfrac43\pi\cdot 9^3={972\pi}$ cubic meters. Let's denote the total weight of the air as $ w$. Now we can plug ${\text{density}=0.9}$, ${\text{total quantity}=w}$, and ${\text{volume}=972\pi}$ in the equation. $\begin{aligned} {\text{Density}}&=\dfrac{{\text{Total quantity}}}{{\text{Volume}}} \\\\ {0.9}&=\dfrac{{w}}{{972\pi}} \\\\ {972\pi}\cdot{0.9}&=\dfrac{{w}}{\cancel{{972\pi}}}\cdot\cancel{{972\pi}} \\\\ 874.8\pi&= w \\\\ 2748&\approx w \end{aligned}$ Ariah needs approximately $2748$ kilograms of air to fill the balloon.